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Consider a smooth projective threefold $\overline W$, constructed in section 4 of this paper. This threefold is a resolution of singularities of the quotient of a product of a K3 surface and $\mathbb CP^1$ by $(\rho, \psi)$, where $\rho$ is a non-symplectic, non-fixed-point-free involution on the K3 surface and $\psi$ is an involution on $\mathbb CP^1$ (the paper (sec 4) contains more details about the construction).

Then $\overline W$ is simply-connected and it has a smooth K3 surface in its anticanonical system.

Here is my question:

Is $\overline W$ irrational? How can one show it?

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    $\begingroup$ It does not look easy to understand the definition in the paper you cite, so it would be better if you provide more details here. $\endgroup$
    – Sasha
    Commented Dec 29, 2021 at 7:25
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    $\begingroup$ If the quotient of the K3 surface is irrational, then you are done (but I guess the construction -- whatever it is -- precludes this case). If the quotient is rational, then your threefold fibers over that rational surface with geometric fibers that are rational curves. Since the involution on the projective line also has fixed points, this projection to the rational surface has a section. For a threefold that is a conic bundle over a rational surface and admits a section, the threefold is rational (in fact the base change to the generic point of the surface is rational). $\endgroup$
    – Jason Starr
    Commented Dec 29, 2021 at 11:49

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I am just posting my comment as an answer. All such threefolds are rational.

By the hypotheses on the involution of the K3 surface, the quotient surface is a rational surface. The projection from the threefold to the rational surface has geometric generic fiber isomorphic to $\mathbb{P}^1$, the other factor in the product. Every fixed point of the involution on $\mathbb{P}^1$ gives a section of this projection.

A normal projective variety that admits a surjective morphism to a target variety is birational (over that target variety) to a product of the target variety and $\mathbb{P}^1$ if and only if the geometric generic fiber is isomorphic to $\mathbb{P}^1$ and there exists a section. Therefore, the threefold is birational to a product of the quotient rational surface and $\mathbb{P}^1$, i.e., it is rational.

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